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Hidden MRF
y 1 =1
y 2 =1
Must-link ( c
=1)
12
Cannot−link
( c
= 1)
23
Must-link ( c
14
=1)
y 5 =3
y 4 =1
y 3 =2
x 1
x 2
x 4
x 5
x 3
Observed data
FIGURE 7.8 : A hidden Markov random field.
C
Θ
Y
X
FIGURE 7.9 : Graphical plate model of variable dependence.
model can be factorized as follows:
P( X, Y, Θ
|
C )=P(Θ
|
C )P( Y
|
Θ ,C )P( X
|
Y, Θ ,C )
(7.15)
The graphical plate model (10) of the dependence between the random
variables in the HMRF is shown in Figure 7.9. The prior probability of Θ is
assumed to be independent of C ,sothatP(Θ
C ) = P(Θ). The probability of
observing the label configuration Y depends on the constraints C and current
generative model parameters Θ. Observed data instances corresponding to
variables X are generated using the model parameters Θ based on cluster
labels Y , independent of the constraints C .Thevariables X are assumed to
be mutually independent: each x i is generated individually from a conditional
probability distribution P( x i |
|
y i , Θ).
Basu et al. (6) show that the joint probability on the HMRF is equivalent
to maximizing:
C )=P(Θ) 1
v ( i, j ) n
y i , Θ)
Z exp
P( X, Y, Θ
|
p ( x i |
(7.16)
c ij ∈C
i =1
They chose the following Gibbs potential for P( Y
|
Θ ,C ):
 
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